the string topology of holomorphic curves in bu n a ...td547cq7455/thesis october 3… · while...

THE STRING TOPOLOGY OF HOLOMORPHIC CURVES IN BU (N)

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF MATHEMATICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Sam Nolen

December 2015

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/td547cq7455

© 2015 by Samuel Richard Nolen. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/td547cq7455

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Ralph Cohen, Primary Adviser


Gunnar Carlsson


Soren Galatius

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Acknowledgements

I owe a great deal to my advisor, Ralph Cohen, who proposed the subject of this thesis. Ralph was exceedingly

generous with his time, expertise, and wisdom during my time in graduate school and helped me navigate

many obstacles. Gunnar Carlsson was also a source of sage advice.

While writing this thesis, I also benefited from conversations with Soren Galatius, Elizabeth Gasparim,

Sadok Kallel, Alexander Kupers, Cary Malkiewich, Jeremy Miller, Sam Nariman, John Pardon, and Jenny

Wilson.

I am very grateful to my parents, Tim and Sharon Nolen, and my sister, Abby, who were extremely

supportive throughout my pursuit of a Ph.D. My cat Lentil deserves a special thanks for frequent vocal con-

tributions. Finally, I would not have completed this thesis without Lauren’s love, patience, and unwavering

belief in me.

Preface

In this thesis, we study spaces Hol(Gr(n ,m)) of holomorphic maps CP1 → Gr(n ,m), where Gr(n ,m)

is the complex Grassmannian of complex n-planes in Cn+m . These holomorphic mapping spaces are related

to many classical areas of mathematics. In particular, their homology was completely determined by Mann

and Milgram, who were motivated by questions in control theory. However, we will study these holomorphic

mapping spaces from a novel viewpoint, the viewpoint of string topology.

“Classical” string topology constructs operations on the the homology of LM Map(S1 ,M), where M

is a smooth manifold; in particular the “loop product”

Hi (LM) ⊗ H j (M) −→ Hi+ j−d (LM).

Gruher and Salvatore extended the construction of the loop product to fiberwise monoids over M. Examples

of such objects include Map(Sn ,M) for arbitrary n.

In this thesis we begin by showing that Hol(Gr(n ,m)) has the structure of a fiberwise monoid over

Gr(n ,m). (This was essentially proved by Kallel and Salvatore in [18], although they used different lan-

guage.) Therefore, there is a “loop product” on the level of homology

Hi (Hol(Gr(n ,m)) ⊗ H j (Hol(Gr(n ,m)) −→ Hi+ j−d (Hol(Gr(n ,m))

These operations were completely computed in the case n 1 (i.e. for Hol(CPn )) by Kallel and Salva-

tore in [18], leaving the question open for n > 1. The only existing result for n > 1 is a complete computation

in the case n 2,m 2, due to Hammouda in [16]. Hammouda’s computation, even though it is a small,

special case, is rather complicated (he computes all the differentials in a spectral sequence with non-zero

differentials on the E2 , E4, and E6 pages), so one expects the general case to be quite difficult.

The main innovation of this thesis is that this difficult problem becomes much easier when Hol(Gr(n ,m))

is replaced by a “stabilized holomorphic mapping space” Holst (Gr(n ,m)), which is equipped with a map to

Hol(Gr(n ,m)). We are able to completely determine the string topology operations on Holst (Gr(n ,m));

in particular, this gives a computation of string topology operations on the union, which we call Holst (BU (n)).

This is what is meant by the string topology of holomorphic curves in BU (n).

Contents

Preface

1 Background and preliminaries 1

1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 String topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.3 Generalized string topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.4 String topology pro-spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 En-ring spectra from fiberwise Cn-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 String topology of holomorphic curves in BU (n) 13

2.1 Fiberwise monoid structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Fiberwise C2-space structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.2 E2-ring spectrum structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 A Cohen-Jones-Yan-type spectral sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 The main calculation, part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 The main calculation, part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Cohomology, tableaux, and homology operations 33

3.1 Cohomology of holomorphic curves in BU (n) . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.1 Young tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.2 Maps in the Cohomology Ind-System . . . . . . . . . . . . . . . . . . . . . . . . . 35

Bibliography 37

Chapter 1

Background and preliminaries

The subject of string topology was founded by Chas and Sullivan in [10], in which they described the structure

of a BV-algebra on H∗(LM), where M is a smooth manifold of dimension d, and LM denotes the free loop

space. In particular, they gave a geometric description of a chain-level intersection or string product

Ci (LM) × C j (LM) −→ Ci+ j−d (LM)

which induces a product on homology.

String topology in its modern, homotopy-theoretic guise was introduced in [6]. This paper introduced a

ring spectrum LM−TM such that a Thom isomorphism induces a commutative square

Hq (LM−TM ) ⊗ Hr (LM−TM ) //

Hq+r (LM−TM )

Hq+d (LM) ⊗ Hr+d (LM) // Hq+r+d (LM),

where the top map is induced by the ring spectrum structure of LM−TM and the bottom map is the Chas-

Sullivan product.

Our main theorem is a computation which combines several of the extensions of the foundations of

string topology, namely, the definition of string topology of classifying spaces via fiberwise monoids due to

Gruher-Salvatore, and the definition of string topology in the setting of holomorphic mapping spaces due to

Kallel-Salvatore.

1

CHAPTER 1. BACKGROUND AND PRELIMINARIES 2

Our main theorem is stated as follows, although its terms will need to be precisely defined.

Theorem 1.0.1. There is an isomorphism of graded rings

H∗(

Holst (BU (n))−TBU (n))

(H∗(Rat(BU (n)))) [[c1 , . . . , cn]]

where the left side is equipped with the Cohen-Jones product, and the right side is a power series ring over

H∗(Rat(BU (n))), which is equipped with the Pontrjagin product.

This in fact gives an explicit description of the left-hand side in terms of generators and relations, in view

of the theorem of Cohen, Lupercio, and Segal ([9], Theorem 2) which identifies the space of degree k based

holomorphic maps Ratk (BU (n)) with the kth piece Fk ,n of the Mitchell-Richter filtration of H∗(ΩSU (n)).

This chapter defines the main players in this story. We give the definition of the generalized string topol-

ogy spectrum associated to a fiberwise monoid over a closed manifold, due to Gruher and Salvatore, and use

this to construct a pro-ring spectrum of holomorphic maps CP1 → BU (n). Finally, we define the string

topology of holomorphic curves in BU (n) as the limit of the result pro-system of homology algebras.

Chapter 2 will be devoted to the proof of the main theorem, and Chapter 3 will contain additional infor-

mation about the cohomology of Holst (BU (n))−TBU (n) and homology operations.

1.1 Preliminaries

1.1.1 Notation

Throughout this thesis, We write Gr(n ,m) to denote the Grassmannian of complex n-planes in Cn+m , and

BU (n) will denote the classifying space of the unitary group U (n).

We will write im for the canonical inclusion

im : Gr(n ,m) → BU (n)

By Rat(Gr(n ,m)) we mean based holomorphic maps from CP1 to Gr(n ,m), and Ratk (Gr(n ,m))

denotes the degree k component of Rat(Gr(n ,m)). By Rat(BU (n)) we mean limm Rat(Gr(n ,m)), and


this is topologized as a subspace of

C∞(

S2 , BU (n))

limm

C∞(

S2 ,Gr(n ,m))

where each C∞(S2 ,Gr(n ,m)) is equipped with the compact-open topology. By Hol(Gr(n ,m)) we mean

unbased holomorphic maps from CP1 to Gr(n ,m), and again Hol(BU (n)) is defined as a limit and topol-

ogized as a subspace of the limit of holomorphic unbased mapping spaces.

It should be noted that Hol(CP1 ,Gr(n ,m)), unlike Hol(CP1 ,CPn ) is not a smooth variety (see [17]).

1.1.2 String topology

In this section we give a quick review of the central construction in string topology as developed by Cohen and

Jones, the spectrum LM−TM . This is relevant because the construction of the “generalized” string topology

spectrum E−TM for any fiberwise monoid E → M is analogous in many ways.

The key technical ingredient is the following generalization of the classical Pontrjagin-Thom construction

due to Cohen-Jones.

Theorem 1.1.1. (Cohen-Jones, [6]) Suppose that f : M → M′ is a smooth embedding and the following

diagram is a homotopy pullback.

E //

p

E′

p′

Mf

// M′

Then there is a Pontrjagin-Thom collapse map

X −→ E(p∗ν) ,

where ν is the normal bundle of M in M′.

The construction of the spectrum LM−TM is as follows. Suppose M is a smooth, closed, oriented mani-

fold of dimension d. Then consider the pullback bundle

LM ×M LM //

LM × LM

ev×ev

M∆

// M ×M


Theorem 1.1.1 gives a Pontrjagin-Thom collapse map

τ : LM × LM −→ (LM ×M LM)ev∗ (ν)

and since ν TM as bundles over M, this is actually a map

LM × LM −→ (LM ×M LM)TM ,

where by abuse of notation we write TM for ev∗(TM).

Twisting the above map by −TM ⊕ −TM and using the loop concatenation m : LM ×M LM → LM

then yields the ring spectrum structure:

LM−TM ∧ LM−TM −→ (LM ×M LM)−TM −→ LM−TM

Theorem 1.1.2. (Cohen-Jones) Let M be a smooth, closed manifold of dimension d. The spectrum LM−TM

is a ring spectrum with unit, whose ring structure is compatible with the Chas-Sullivan product, in the sense

that the following diagram commutes.

Hq (LM−TM ) ⊗ Hr (LM−TM ) //

Hq+r (LM−TM ∧ LM−TM ) // Hq+r (LM−TM )

Hq+d (LM) ⊗ Hr+d (LM) // Hq+r+d (LM)

1.1.3 Generalized string topology

Gruher and Salvatore developed string topology in the setting of fiberwise monoids.

Definition 1.1.3. A fiberwise monoid is a fiber bundle p : E → B equipped with a fiberwise map

m : E ×B E −→ E

and a section

s : B → E


satisfying

m(x , s(p(x))) m(s(p(x)), x) x

and

m(m(x , y), z) m(x ,m(y , z))

A map of fiberwise monoids is simply a bundle map

E //

E′

B // B′

However, in this thesis the maps of fiberwise monoids which we consider will be slightly less general

than this. All maps we consider will have one of the two following properties: either 1) the map B → B′ will

be a homeomorphism or 2) the map E → E′ will be a pullback along the map B → B′.

Proposition 1.1.4. Suppose p : E → B is a fiberwise monoid and let f : B′ → B. Then f ∗E → B′ naturally

has the structure of a fiberwise monoid.

Theorem 1.1.5. [13], 4.3.2 Let

F −→ E −→ M

be a smooth fiberwise monoid over a closed finite-dimensional manifold manifold. Then the Thom spectrum

E−TM is an associative ring spectrum with unit, and the induced map

p : E−TM −→ M−TM

is a map of ring spectra. In the case of the fiberwise monoid

ΩM −→ LM −→ M,

this construction agrees with the ring spectrum LM−TM .

The construction essentially mimics the case of E LM. Applying the Pontrjagin-Thom construction

Theorem 1.1.1 to the square


E ×M E //

E × E

M // M ×M

yields a map

E × E −→ (E × E)TM .

Twisting by −TM ⊕ −TM and composing with the fiberwise multiplication m : E ×M E → E then yields

the ring spectrum structure:

E−TM ∧ E−TM −→ (E ×M E)−TM −→ E−TM

The following is the key example. Let M is a smooth manifold and p ∈ Sn . Then consider the evaluation

map evp : Maps(Sn ,M) → M, given by φ 7→ φ(p).

One would like to define the fiberwise multiplication m as follows.

Maps(Sn ,M) ×M Maps(Sn ,M) //

Maps(Sn ∨ Sn ,M) // Maps(Sn ,M)

M // M

and the section s : M → Maps(Sn ,M) be given by

s(m)(q) m for all m ∈ M, q ∈ Sn .

This doesn’t quite work because this multiplication is only associative up to homotopy. However, we can

fix this by using a version of the classic “Moore loops” construction.

Fiberwise monoids have the following functoriality properties, which follow from Cohen and Jones’

generalized Pontrjagin-Thom construction Theorem 1.1.1.

Proposition 1.1.6. ([14], Theorem 8, 1). Let p : E → M be a fiberwise monoid with fiber F, and f : M′ →

M a smooth map of closed manifolds. Consider the pullback fibration p′ : f ∗E → M′ defined by the pullback

square


f ∗E //

p′

E

p

M′

f// M

Then p′ : f ∗E → M′ naturally has the structure of a fiberwise monoid with fiber F and f induces a ring

spectrum map

φ f : E−TM −→ f ∗E−TM′

Proposition 1.1.7. ([14], Theorem 8, 2). Let p : E → M and p′ : E′ → M be fiberwise monoids with fiber

F and F′, respectively, and suppose we are given a map of fiberwise monoids

F //

F′

E //

p

E′

p′

M

// M

Then p′ : f ∗E → M′ naturally has the structure of a fiberwise monoid with fiber F and f induces a ring

spectrum map

ψ f : E−TM −→ E′−TM

1.1.4 String topology pro-spectra

In this section we will define the main objects under consideration: string topology pro-spectra and stable

string topology spectra.

First we will consider pro-objects in fiberwise monoids over manifolds. In other words, we have a system

E• of fiberwise monoids

Fk −→ Ek

pk−→ Mk for k ∈ N

and for any j < k we have morphisms of fiberwise monoids


F j//

Fk

E j//

Ek

M j// Mk

satisfying the obvious compatibility condition.

We can associate a “stable” pro-fiberwise monoid (Est )• as follows. Note that

hocolim F j −→ hocolim E j −→ hocolim M j

naturally has the structure of a fiberwise monoid. We define (Est )• to be the fiberwise monoid

hocolim F j −→ i∗k

(

hocolim E j

)

−→ Mk ,

where ik is the natural morphism Mk → hocolim M j .

Note that there is a pullback of fiberwise monoids

i∗k

(

hocolim E j

)//

ik+1

(

hocolim E j

)

Mk// Mk+1

So, by Proposition 1.1.7, there are maps of spectra

i∗k+1

(

hocolim E j

)−TMk+1−→ i∗k

(

hocolim E j

)−TMk

and there is a resulting pro-ring spectrum which we denote by ((Est )•)−Thocolim M j .

Definition 1.1.8. We will denote the pullback fiberwise monoid

i∗mHol(BU (n)) −→ Gr(n ,m)

by Holst (Gr(n ,m)).


By Holst (BU (n))−TBU (n) , we will mean the pro-ring spectrum associated to the stable pro-fiberwise

monoid Holst (BU (n)) obtained from the pro-fiberwise monoid

· · · // Hol(Gr(n ,m)) //

Hol(Gr(n ,m + 1)) //

· · ·

· · · // Gr(n ,m)) // Gr(n ,m + 1) // · · ·

Care is necessary here because we must distinguish between maps

H∗(i∗m+1Hol(BU (n))−TGr(n ,m+1)) −→ H∗(i∗m (Hol(BU (n)))−TGr(n ,m))

induced from spectrum maps, and maps

H∗(i∗m+1(Hol(BU (n)))) −→ H∗(i∗m (Hol(BU (n)))),

because the latter are maps of algebras, while the former are not. These maps of algebras shift the degree by

2n. In fact, the following diagram commutes, where the vertical maps are Thom isomorphisms.

H∗(Hol(BU (n))−TGr(n ,m+1)) //

H∗(Hol(BU (n))−TGr(n ,m))

H∗+2n(m+1) (Hol(BU (n)))∩e (ν)

// H∗+2nm (Hol(BU (n)))

where the bottom map is given by the cap product with the Euler class of the normal bundle of Gr(n ,m) in

Gr(n ,m + 1). This follows by naturality from the following proposition.

Proposition 1.1.9. Let i : Nn → Mm and consider the induced map of Thom spectra

τ(i) : M−TM −→ N−TN .

Then there is a commutative diagram

Hq (N−TN )(τ(i))∗

//

Hq (M−TM )

Hq+n (N)i! // Hq+m (M)

i∗// Hq+m (N)

where the vertical maps are Thom isomorphisms, and the composition i∗i! is given by taking the cup product

with e (ν(i)), the Euler class of the normal bundle of i.


Proof. The multiplicativity of the Thom class gives

Th(TM |N ) Th(TN) ∪ Th(ν)

On the other hand, under the Thom isomorphism, the Thom class Th(ν) corresponds to the Euler class e (ν).

1.2 En-ring spectra from fiberwise Cn-spaces

In what follows, C2 denotes the little 2-disks operad. So C2(k) is the space of affine embeddings∐k

i1 U2 →

U2, where U is an open disk in R2. For a definition in full detail, see May’s book [20], although it works

with cubes instead of disks.

In this section we review the following theorem of Gruher and Salvatore.

Theorem 1.2.1 (Gruher-Salvatore, [14], Theorem 13). Suppose Ef−→ M is a fiber bundle and a fiberwise

Cn-space. Then E−TM is an En-ring spectrum.

We will need not only this theorem but the details of their construction, in order to prove Theorem 1.2.3

below. The construction proceeds as follows. Let ∆k (M)ik→ Mk denote the “thin diagonal”, i.e. the set of

k-tuples of pairwise distinct elements of M topologized as a subspace of the product. On the one hand, we

have structure maps

αk : Cn (k) × (πk )−1(∆k (M)) −→ E

which yield maps of Thom spectra

α−TMk

: Σ∞Cn (k)+ ∧ (πk )−1(∆k (M))−TM −→ E−TM

On the other hand, applying the generalized Pontrjagin-Thom construction Theorem 1.1.1 to the pullback

diagram

(πk )−1(∆k (M)) //

Ek

πk

∆k (M)ik // Mk


yields a spectrum map

φk : (E−TM )∧k −→(

πk)−1

(∆k (M))−TM

Then the En-ring structure on E−TM is given by the composition

Σ∞Cn (k)+ ∧ (E−TM )k −→ Σ∞Cn (k)+ ∧ (πk )−1(∆k (M))−TM −→ E−TM

Definition 1.2.2. A homotopy morphism of En-ring spectra E∼→ E′ is a map of spectra f : E → E′ together

with a homotopy commutative diagram of operads

En

##

End(E) // End(E′)

Theorem 1.2.3. Suppose we have a diagram of fiberwise monoids

F //

F′

E //

E′

M // M

such that E and E′ are also equipped with the structure of fiberwise Cn-spaces, but this structure is preserved

in the above diagram only up to homotopy. Then the resulting map of spectra

E−TM −→ (E′)−TM

is a homotopy morphism of En-spectra.

Proof. What we need to show is that the following diagram commutes up to homotopy.

Σ∞C2(k)+ ∧ (E−TM )∧k //

Σ∞C2(k)+ ∧ ((E′)−TM )∧k

Σ∞C2(k)+ ∧ (πk )−1 (∆k (M))−TM //

Σ∞C2(k)+ ∧ (πk )−1 (∆k (M))−TM

E−TM // (E′)−TM


The bottom square commutes up to homotopy by hypothesis. The top square commutes for the following

reason. Since the map E → E′ is a map of fiberwise monoids over M, the following diagram commutes up

to homotopy.

(πk )−1(∆k (M)) //

''

((π′)k )−1(∆k (M))

ww

∆k (M)

Ek //

''

(E′)k

wwMk

This proves the theorem.

Chapter 2

String topology of holomorphic curves in

BU (n)

In this chapter we will carry out our main computation, Theorem 1.0.1. The plan of this chapter is as follows.

In section 2.1 we will prove that Hol(Gr(n ,m)) → Gr(n ,m) is a fiberwise C2-algebra. In Section 2.2

we derive a second-quadrant spectral sequence computing the algebra H∗+d (E−TM ) for E → M a fiberwise

monoid. In Section 2.3 and 2.4 we use the spectral sequence to prove Theorem 1.0.1.

2.1 Fiberwise monoid structure

The goal of this section is to prove the following:

Proposition 2.1.1. The evaluation map ev∗ : Hol(Gr(n ,m)) → Gr(n ,m) is a fiberwise C2-space with

fiber Rat(Gr(n ,m)).

We choose to work primarily with the specific case of Gr(n ,m) rather than following precedent by

working with the general case of generalized flag manifolds. For clarity, we review the definition:

Definition 2.1.2. Let G be a semisimple algebraic group. A subgroup B ⊂ G is said to be a Borel subgroup

if it is a maximal Zariski closed and connected solvable algebraic subgroup. A subgroup P ⊆ G is called a

13

CHAPTER 2. STRING TOPOLOGY OF HOLOMORPHIC CURVES IN BU (N) 14

parabolic subgroup if it contains some Borel subgroup. If P is a parabolic subgroup, then G/P is said to be

a generalized flag manifold.

Several authors have defined a C2-action on Rat(G/P), where G/P is a generalized flag manifold; cf.

Boyer-Hurtubise-Mann-Milgram in [3], Cohen-Jones-Segal in [7], and Valli in [26]. We will give the explicit

details of one such action (due to Hammouda in [16]) in the case of interest to us, G/P Gr(n ,m). The

goal will be to show that this C2-action on the fiber of Hol(Gr(n ,m)) → Gr(n ,m) extends to the structure

of a fiberwise C2-algebra.

Moreover, it should be noted that, in [18], the construction of the ring spectrum Hol(G/P)−T (G/P) for

generalized flag manifolds G/P is already given by Kallel and Salvatore. However, we will need not just

the structure of the spectrum but also the fiberwise monoid structure in order to establish the functoriality

properties given in Propositions 1.1.6 and 1.1.7.

2.1.1 Fiberwise C2-space structure

We begin this section with an exposition of the convenient C2-algebra structure on Rat(Gr(n ,m)) given by

Hammouda in [16]. We will use the following theorem of Mann and Milgram:

Theorem 2.1.3. (Mann-Milgram, [21]) Any f ∈ Ratk (Gr(n ,m)) has a unique representation

[D |N]

p11(z) p12(z) · · · p1n (z) q11(z) · · · q1m (z)

0 p22(z) · · · p2n (z) q21(z) · · · q2m (z)

......

. . ....

......

...

0 0 · · · pnn (z) qn1(z) · · · qnm (z)

satisfying the following conditions:

1) For each i, pii (z) is monic.

2) For each j < i, deg p ji < deg pii .

3) For each j, deg qn j < deg pnn .

For f ∈ Rat(Gr(n ,m)) with Mann-Milgram normal form [D |N], consider the “transfer matrix” D−1(z)N (z),


which is a n × m matrix of rational functions which we rewrite in the form

T f (z) ∑

i , j

Ai j

(z − λi ) j,

where λi are the roots of det(D(z)), and Ai j ∈ Matn×m (C).

We identify any open disk

U (r, x0) z ∈ C : |z − x0 | < r ,

with C by the map

φU (r,x0) : ρe iθ 7→2r

πarctan (ρ)e iθ + x0.

Let p(z) a(z − λn )(z − λn−1) · · · (z − λ0). Then given an open disk U U (r, x0), we associate a

map

pU (z) a(z − φU (λn )) · · · (z − φU (λ0))

Then if

f (z) ∑

i , j

Ai j

(z − λi ) j

is a rational function with coefficients in Matn×m (C), we write

fU (z) ∑

i , j

Ai j

(z − φU (λi )) j


Above: locations of poles for α3(U1 ,U2 ,U3 , f , f , f ), where f has two poles.

Then we define C2-algebra structure maps on Rat(Gr(n ,m)) by

αk : C2(k) × Rat j1 (Gr(n ,m)) × · · · × Rat jk(Gr(n ,m)) −→ Rat∑

i ji (Gr(n ,m))

(

Ui , fi )

7→∑

i

( fi )Ui

It is clear that this is invariant under the Σk-action which permutes the fi , and it is associative in the

operadic sense becausek∑

i1

*.,

ji∑

ℓ1

(

f iℓ

)

U iℓ

+/-Ui

∑

i ,ℓ

(

f iℓ

)

φUi(U i

ℓ)

So this defines a C2-structure on Rat(Gr(n ,m)).

Example. Let

f (z)

z 0 0 −1

0 z2 + 1 1 0

,


so that f has associated transfer matrix

T f (z)

0 − 1z

1z2+1

0

1

z

0 −1

0 0

+

1

z − i

0 0

1 0

+

1

z + i

0 0

1 0

Let U1 be the disk in C with center −1 + i and radius 1, and let U2 be the disk in C with center 1 and radius

2. Then

α1(U1 , F) 1

z + 1 − i

0 −1

0 0

+

1

z + 1 − 3i2

0 0

1 0

+

1

z + 1 − i2

0 0

1 0

α1(U2 , F) 1

z − 1

0 −1

0 0

+

1

z

0 0

1 0

+

1

z − 2

0 0

1 0

α2((U1 ,U2), (F, F))

0 2z−iz2−iz−1+i

16z3−24iz2−(15−8i)z−(1−8i)(z−2)z(2z+2−i)(2z+2−3i) 0

Proposition 2.1.4. The inclusion

jm : Rat(Gr(n ,m)) −→ Rat(Gr(n ,m + 1))

respects the above C2-structure.

Proof. Suppose that jm ( f ) f ′. Then the Mann-Milgram normal form [D |N] of f is obtained from the

Mann-Milgram normal form [D |N′] by removing the right-most column. So if we have

D−1(z)N′(z) ∑

i , j

Ai j

(z − λi ) j,

then we also have

D−1(z)N′(z) ∑

i , j

A′i j

(z − λi ) j,

where Ai j is obtained from A′i j

by removing the right-most column.


2.1.2 E2-ring spectrum structure

We will be using the following fact pointed out by Kallel and Salvatore in [18]:

Proposition 2.1.5. Let G/P be a generalized flag manifold. The map

G × Rat(G/P) −→ Hol(G/P) : (g · φ)(z) g · φ(z)

induces a homeomorphism

Hol(G/P) G ×P Rat(G/P)

In particular, this implies the following.

Proposition 2.1.6. If G/P is a generalized complex flag manifold, the evaluation map

ev : Hol(G/P) −→ G/P : φ 7→ φ(∗)

is a fiber bundle.

This is stronger than the result of Cohen, Jones, and Segal in [7] that

ev∞ : Rat(G/P) −→ G/P,

is a quasifibration. It is important for our purposes that the evaluation map be a Serre fibration rather than just

a quasifibration because quasifibrations are not closed under pullbacks (Dold-Thom [11], Bemerkung 2.3).

Proof of 2.1.1. It is important to note that the C2-structure defined above on Rat(Gr(n ,m)) is U (n) ×

U (m)-equivariant. The reason for this is simply that, if B ∈ U (n) ×U (m) and Ui ∈ C2(k),

B*.,

k∑

ℓ1

∑

i , j

Ai j

(z − φUℓ (λi )) j

+/-

k∑

ℓ1

∑

i , j

BAi j

(z − φUℓ (λi )) j

Now, let

αk : C2(k) × Rat j1 (Gr(n ,m)) × · · · × Rat jk(Gr(n ,m)) −→ Rat∑k

i1ji

(Gr(n ,m))


be the C2-structure maps for Rat(Gr(n ,m)).

Using the U (n)×U (m)-equivariance of the C2-action, we define fiberwise C2-structure maps on Hol(Gr(n ,m))

over Gr(n ,m) to be the composition

C2(k) × Hol j1 (Gr(n ,m)) ×Gr(n ,m) · · · ×Gr(n ,m) Hol jk(Gr(n ,m)

C2(k) × *,U (n + m) ×U (n)×U (m)

*,

k∏

i1

Rat ji (Gr(n ,m))+-

+-

U (n + m) ×U (n)×U (m)*,C2(k) × *

,

k∏

i1

Rat ji (Gr(n ,m))+-

+-

U (n+m)×U (n)×U (m)αk−→ U (n + m) ×U (n)×U (m) Rat∑k

i1ji

(Gr(n ,m)) Hol∑ki1

ji(Gr(n ,m))

This defines a C2-algebra structure because the αk define a C2-algebra structure.

Corollary 2.1.7. Let G/P be a generalized complex flag manifold. The ring spectrum Hol(G/P)−T (G/P)

defined in Kallel-Salvatore is naturally an E2-ring spectrum.

Proof. Theorem 2.1.1 tells us that, if G/P is a generalized flag manifold, then

Rat(G/P) −→ Hol(G/P) −→ G/P

is a fiberwise C2-space. Moreover, the construction of Gruher-Salvatore CITE tells us that if E → M is

a fiberwise Cn-space, then E−TM is an En-ring spectrum. it follows that Hol(G/P)−T (G/P) is an E2-ring

spectrum.

We should note that the following was stated for the case n 1 by Boyer and Mann in [4], Theorem 4.16.

Related results were proven by Valli in [26] and Cohen, Jones, and Segal in [7].

Theorem 2.1.8. The C2-actions on Rat(Gr(n ,m)) and on Ω2(Gr(n ,m)) are homotopy compatible. In

other words, the following diagram commutes up to homotopy for all k.

C2(k) ×Σk(Rat(Gr(n ,m)))k //

Rat(Gr(n ,m))

C2(k) ×Σk

(

Ω2(Gr(n ,m))

) k// Ω

2(Gr(n ,m))


Proof. It suffices to provide homotopies

Ai

(z − φ(λi )) j∼

Ai

(φ(z−λi )) j if z ∈ Ui

∞ otherwise

Using polar coordinates z ρe iθ , λi ρ′e iθ′ , we have

z − φ(λi )

(

ρ cos θ −2r

πarctan(ρ′) cos θ′

)

+ i(

ρ sin θ −2r

πarctan ρ′ sin θ′

)

φ (z − λi )

2r

πarctan

((

(ρ cos θ − ρ′ cos θ′)2 + (ρ sin θ − ρ′ sin θ′)2) 1

2

)

(

(ρ cos θ − ρ′ cos θ′) + i(ρ sin θ′ − ρ′ sin θ′))

For 0 ≤ t ≤ 1, let

htr,x0

: ρe iθ 7→

(

(1 − t)ρ +2rt

πarctan(ρ)

)

e iθ + tx0

H tr,x0 ,λi

(z) :

z − h1−2tr,x0

(λi ) if 0 ≤ t ≤ 12

h12+ 1

2t

r,x0(z − λi ) if 1

2≤ t ≤ 1

Then the homotopy we need is

t 7→Ai

H tr,x0 ,λi

(z)

Theorem 2.1.9. The inclusion

Hol(Gr(n ,m)) −→ Map(S2 ,Gr(n ,m))

induces a spectrum map

Hol(Gr(n ,m))−TGr(n ,m) −→ Map(S2 ,Gr(n ,m))−TGr(n ,m)


which is a map of E2-ring spectra up to homotopy.

Proof. This is immediate from Theorem 2.1.8, using Theorem 1.2.3.

2.2 A Cohen-Jones-Yan-type spectral sequence

In this section, we derive a spectral sequence for computing the homology of the generalized string topology

spectrum E−TM for E → M a fiberwise monoid, and we show that it collapses at the E2 page in our case of

interest.

In [8], Cohen, Jones, and Yan developed a spectral sequence for computing the algebraH∗(LM) equipped

with the loop product was developed, and used to compute H∗(LSn ) and H∗(LCPn ) as algebras. The main

theorem of that paper is the following.

Theorem 2.2.1. Let M be a closed, oriented manifold. There is a second quadrant spectral sequence of

algebras Erp ,q such that

1) Er∗,∗ is an algebra and the differential

dr : Er∗,∗ −→ Er

∗−r,∗+r−1

is a derivation for each r ≥ 1.

2) The spectral sequence converges to the loop homology H∗(LM) as algebras. In other words, E∞∗,∗ is

the associated graded algebra for a natural filtration of the algebra H∗(LM).

3) For m , n ≥ 0, there is an isomorphism

E2−m ,n Hm (M; Hn (ΩM)).

of algebras, where the algebra structure on H∗(M; H∗(ΩM)) is the cup product on the cohomology of M

with coefficients in the Pontrjagin ring H∗(ΩM).

We will generalize this as follows. It should be noted that Gruher states this as her Theorem 4.3.6 in [13]

but does not prove it. This includes Theorem 2.2.1 as a special case when the fiberwise monoid is LM → M,


and it includes Proposition 6.1 of [18] as a special case when the fiberwise monoid is Map(Sn ,M) → M.

We follow the strategy of the proof of Proposition 6.1 in [18].

Theorem 2.2.2. Let F → E → M be a fiberwise monoid with M a smooth, closed, oriented, simply

connected manifold. Suppose also that

E ×M E ⊆ E × E

has a tubular neighborhood homeomorphic to the pullback of the tangent bundle along the fibration E → M.

Then there is a second quadrant spectral sequence of algebras Erp ,q , d

r : p ≤ 0, q ≥ 0 such that

1) Er∗,∗ is an algebra and the differential dr : Er

∗,∗ → Er∗−r,∗+r−1

is a derivation for each r ≥ 1.

2) The spectral sequence converges to H∗(E) as algebras.

3) For m , n ≥ 0,

E2−m ,n Hm (M; Hn (F)),

and the isomorphism E2−∗,∗ H∗(M; H∗(F)) is an isomorphism of algebras, where the algebra structure on

H∗(M; H∗(F)) is given by the cup product on the cohomology of M with values in the Pontrjagin ring H∗(F).

4) If there is a map of fiberwise monoids

F //

F′

E //

E′

M

// M

then there is a map of spectral sequences

φrp ,q : Er

p ,q −→(

E′) r

p ,q

such that

φ2p ,q : H−p (M; Hq (F)) −→ H−p (M; Hq (F′))

is induced by the map of spaces F → F′ and

φ∞p ,q : Hp+q (E−TM ) −→ Hp+q ((E′)−TM )


is induced by the map of spectra obtained from Proposition 1.1.7.

5) If there is a map of manifolds f : M′ → M inducing a pullback of fiberwise monoids

F //

F

f ∗E //

E

M′

f// M

then there is a map of spectral sequences

ψrp ,q : Er

p ,q −→(

f ∗E) r

p ,q

such that

φ2p ,q : H−p (M; Hq (F)) −→ H−p (M′; Hq (F))

is induced by the map of spaces f : M′ → M and

φ∞p ,q : Hp+q (E−TM ) −→ Hp+q (( f ∗E)−TM )

is homology applied to the map of spectra obtained from Proposition 1.1.6.

Proof. Consider the filtration of the singular chain complex

· · · −→ Fp−1C∗(E) −→ FpC∗(E) −→ · · · −→ C∗(E),

where FpC∗(E) is the sub-complex generated by r-simplices σ : ∆r → E such that, for some σ : ∆q → M

with q ≤ p, the following diagram commutes,

∆r σ

//

(i0 ,...,ir )

E

ev

∆q

σ// M

where

(i0 , . . . , ir ) : ∆r −→ ∆q


is defined by sending the kth vertex of ∆r to the ik th vertex of ∆q , with 0 ≤ i0 ≤ · · · ≤ ir ≤ q, and extending

linearly.

Now, F∗C∗(E) is precisely the filtration giving rise to the Serre spectral sequence for the fiber sequence

F → E → M. We want to show existence of the fiberwise multiplication at the chain level

Ci (E) ⊗ C j (E) −→ Ci+ j−d (E)

and respects the filtration

Fp (C∗(E)) ⊗ Fq (E)) −→ Fp+q−d (C∗(E))

We define the fiberwise multiplication as a composition which fits on the left side of the following com-

mutative diagram. Here ν denotes the pullback of TM to E ×M E, as well as the pullback of TM to E, by

abuse of notation.

Ci (E) ⊗ C j (E)

// Ci+ j (E × E)

Ci+ j ((E ×M E)ν)

Ci+ j−d (E) Ci+ j (Eν)oo

The top map is the slant pairing ([25], 13.70). The next map is induced by the Pontrjagin-Thom collapse

map. The next map is induced by the fiberwise multiplication.

The definition of the bottom map in the above diagram is as follows. Let τ ∈ Cd (DM, SM) be a

normalized cochain representative of the Thom class; i.e. a cochain representing the Thom class which

vanishes on degenerate simplices. The map

C∗(Eν) −→ C∗(ev∗(DTM, STM))

is given by excision followed by capping with τ:

C∗(Eν) C∗(ev∗DTM/ev∗STM) −→ C∗(ev∗(DTM), ev∗(STM))

∩τ−→ C∗−d (ev∗DTM)

∼−→ C∗−d (E)


It is necessary to show that capping with τ

FpC∗(ev∗DTM)∩τ−→ Fp−dC∗(ev∗DTM)

decreases filtration degree by d. Let σ ∈ Fp (Cr (DTM)), so that ev∗(σ) σ(i0 , . . . , ir ), i0 ≤ · · · ≤ ir ≤

q ≤ p, and

ev∗(σ ∩ τ) ev∗(σ) ∩ τ σ(i0 , . . . , ir ) ∩ τ

±τ(σ(ir−d , . . . , ir ))σ(i0 , . . . , ir−d ).

Since τ vanishes on degenerate simplices, this last expression is non-zero only if ir−d < . . . < ir ≤ q;

i.e. ir−d ≤ q − d ≤ p − d. If we write (σ |q−d ) for the restriction of σ to its front (q − d)-face, then, for

σ ∈ FpC∗(ev∗DTM),

ev∗(σ ∩ τ) a(σ |q−d )(i0 , . . . , ir−d ),

for some a, so

σ ∩ τ ∈ Fp−dC∗(ev∗DTM),

as desired.

Now, because the intersection product decreases the filtration degree by d, it induces a pairing on each

page of the Serre spectral sequence for the fiberwise monoid E → B,

Erp ,s ⊗ Es

q ,t −→ Erp+q−d ,s+t

Now, this pairing on the E2 page,

Hp (M,Hs (F)) ⊗ Hq (M,Ht (F)) −→ Hp+q−d (M,Hs+t (F))

may be identified with the intersection product with coefficients in the Pontrjagin ring H∗(F). So, the grading-

shifted spectral sequence converges to the generalized string topology ring H∗(E). This proves parts 1-3 of

the theorem.


Now we will address functoriality of the spectral sequence. Suppose we have a morphism of fiberwise

monoids

F //

F′

E //

ev

E′

ev′

M // M

Then we have a chain map

C∗(M; C∗(F)) −→ C∗(M; C∗(F′))

which additively induces a map of spectral sequences. To establish the first functoriality claim, we will show

that the morphism preserves the chain-level intersection product, i.e. that the following diagram commutes.

Ci (E) ⊗ C j (E) //

Ci (E′) ⊗ C j (E

′)

Ci+ j (Eν) //

Ci+ j ((E′)ν)

Ci+ j−d (E) // Ci+ j−d (E′)

The top square commutes because it is obtained from the the following commutative diagram of spaces.

E × E //

E′ × E′

(E ×M E)ν //

(E′ ×M E′)ν

Eν // (E′)ν

The commutativity of the bottom square

Ci+ j (Eν)

f∗//

∩τE

Ci+ j ((E′)ν)

∩τE′

Ci+ j−d (E)f∗

// Ci+ j−d (E′)

is a consequence of the fact that f ev′ ev, so that ev∗ (ev′)∗ f ∗ and f∗τE f∗ f ∗τE′ τE′ .

Finally, suppose that we have a pullback of fiberwise monoids


F≃ //

F

f ∗E //

ev′

E

ev

M′ // M

Then we have a chain map

C∗(M; C∗(F)) −→ C∗(M′; C∗(F))

which we want to show induces the Pontrjagin-Thom collapse map in homology on the E∞-page. As before,

the following square commutes because it arises from a map of spaces.

Ci (E) ⊗ C j (E) //

Ci ( f ∗E) ⊗ C j ( f ∗E)

Ci+ j (Eν) // Ci+ j (( f ∗E)ν)

The commutativity of the bottom square

Ci+ j (Eν)

f ∗//

∩τE

Ci+ j (( f ∗E)ν)

∩τ f ∗E

Ci+ j−d (E)f ∗

// Ci+ j−d ( f ∗E)

again follows from the fact that f ∗τE τ f ∗E.

2.3 The main calculation, part 1

Lemma 2.3.1. As algebras, H∗(

i∗Hol(BU (n))−TGr(n ,m))

is isomorphic to

H∗(Rat(BU (n)) ⊗ H−∗(Gr(n ,m)),

where H∗(Rat(BU (n))) is equipped with the Pontrjagin product and H−∗(Gr(n ,m)) is equipped with the

cup product.


Proof. For the fiber sequence

Rat(BU (n)) −→ i∗mHol(BU (n)) −→ Gr(n ,m),

the spectral sequence of Theorem 2.2.2 takes the form

H∗(Rat(BU (n))) ⊗ H−∗(Gr(n ,m)) ⇒ H∗(

i∗mHol(BU (n))−TGr(n ,m))

Furthermore, it is a result of Cohen, Lupercio, and Segal in [9] that Ratk (BU (n)) is homotopy equivalent

to the kth stage Fk ,n of the Mitchell filtration of ΩU (n), as defined in [23]. Regarding H∗(ΩU (n)) via its

standard presentation as a polynomial algebra

H∗(ΩU (n)) Z[b1 , . . . , bn−1]; |bi | 2i ,

the homology H∗(Fk ,n ) is spanned by monomials in the bi’s of length at most k.

In particular, this implies that H∗ (Rat(BU (n))) is concentrated in even degrees. Therefore, since

H−∗(Gr(n ,m)) is also concentrated in even degrees (i.e. is zero in odd degrees), this spectral sequence

must degenerate at the E2-page, and there can be no extension issues.

Proposition 2.3.2. The following diagram commutes.

H∗(


//

H∗(

i∗m Map(S2 , BU (n))−TGr(n ,m))

H∗ (Rat(BU (n))) ⊗ H−∗(Gr(n ,m)) // H∗(ΩU (n)) ⊗ H−∗(Gr(n ,m))

under the isomorphism Ω2BU (n) ≃ ΩU (n).

Proof. This follows from functoriality of the spectral sequence applied to the homotopy map of fiberwise

monoids


Rat(Gr(n ,m)) //

Ω2Gr(n ,m)

Hol(Gr(n ,m)) //

Maps(S2 ,Gr(n ,m))

Gr(n ,m) // Gr(n ,m)

given the fact that the spectral sequences for the string topology algebras of both fiberwise monoids collapse.

Theorem 2.3.3. The following diagram commutes:

H∗(

Hol(BU (n))−TGr(n ,m+1))

1⊗Th//

H∗(

Hol(BU (n)))−TGr(n ,m))

H∗(Rat(BU (n))) ⊗ H∗(

Gr(n ,m + 1)−TGr(n ,m+1))

1⊗Th//

1⊗α

H∗(Rat(BU (n))) ⊗ H∗(

Gr(n ,m)−TGr(n ,m))

1⊗α

H∗(Rat(BU (n))) ⊗ H−∗(Gr(n ,m + 1))1⊗i∗

// H∗(Rat(BU (n))) ⊗ H−∗(Gr(n ,m))

Here α is the Atiyah duality isomorphism α : H∗(M−TM ) H−∗(M) and Th is the collapse map induced

by the inclusion Gr(n ,m) → Gr(n ,m + 1) using Proposition 1.1.6.

Proof. The commutativity of the top square follows from functoriality of the spectral sequence, as spelled

out in Theorem 2.2.2. The bottom square is a fact about Atiyah duality (The original reference for this is

Atiyah’s paper [1]).

2.4 The main calculation, part 2

In this section we will prove the main theorem.

Theorem 2.4.1. There is an isomorphism of graded algebras

H∗(

(Holst (BU (n)))−TBU (n))

−→ H∗(Rat(BU (n)))[[c1 , . . . , cn]]

where the left hand-side is equipped with the string product, and the right-hand side is a graded power series

ring, where H∗(Rat(BU (n))) is considered as a ring with the Pontrjagin product. Here |ci | −2i.


We will begin with two necessary algebraic lemmas. We would like to say that

inv limm (H−∗(BU (n); H∗(Rat(BU (n)))) Z[[c1 , . . . , cn]] ⊗ H∗(Rat(BU (n)))

but we will need to be more precise than this. The above are Z-graded rings, with the following gradings:

(H−∗(BU (n); H∗(Rat(BU (n))))k :∏

p+qk

H−p (BU (n); Hq (Rat(BU (n))))

Note that the right-hand side is an infinite product.

We are considering Z[[c1 , . . . , cn]] ⊗ H∗(Rat(BU (n))) a graded ring with the grading

(Z[[c1 , . . . , cn]] ⊗ H∗(Rat(BU (n))))k

∏

p+qk

(Z[[c1 , . . . , cn]])p ⊗ Hq (Rat(BU (n)))

Again, this is an infinite product because, with the grading convention |ci | −2i, Z[[c1 , . . . , cn]] has

nonzero homogenous elements in arbitrarily large negative degrees, while H∗(Rat(BU (n))) has nonzero

homogeneous elements in arbitrarily large positive degrees.

Lemma 2.4.2. The map

inv limm (H−∗(Gr(n ,m); H∗(Rat(BU (n)))) −→ H−∗(BU (n); H∗(Rat(BU (n))))

induced by the system of inclusions Gr(n ,m) → BU (n) for all m, is an isomorphism of graded rings.

Proof. It suffices to work on each graded component. Selecting the kth graded component, our reasoning

proceeds as follows.


(inv limmH−∗(Gr(n ,m); H∗(Rat(BU (n)))))k inv limm

∏

p+qk

H−p(

Gr(n ,m); Hq (Rat(BU (n))))

inv limm

∏

p+qk

(

H−p (Gr(n ,m)) ⊗ Hq (Rat(BU (n))))

∏

p+qk

inv limm

(

H−p (Gr(n ,m)) ⊗ Hq (Rat(BU (n))))

−→∏

p+qk

(

H−p (BU (n)) ⊗ Hq (Rat(BU (n)))

∏

p+qk


(H−∗(BU (n); H∗(Rat(BU (n)))))k

Lemma 2.4.3. The natural map of graded rings

Φ : H∗(Rat(BU (n)))[[c1 , . . . , cn]] −→ H−∗(BU (n); H∗(Rat(BU (n))))

is an isomorphism.

Proof. We have

(H−∗(BU (n); H∗(Rat(BU (n)))))k

∏

p+qk


and

(Z[[c1 , · · · cn]] ⊗ H∗(Rat(BU (n))))k

∏

p+qk

(Z[[c1 , · · · cn]])p ⊗ Hq (Rat(BU (n)))

Φ is grading-preserving and, on the (p , q)th summand, the map Φp ,q fits into the following square of

isomorphisms

(Z[[c1 , · · · cn]])p ⊗ Hq (Rat(BU (n)))

Φp ,q// H−p (BU (n); Hq (Rat(BU (n))))

Hom(H−p (BU (n)),Z) ⊗ Hq (Rat(BU (n))) // Hom(

H−p (BU (n); Hq (Rat(BU (n)))),Z)


where the right-hand map is linear duality, the bottom map is the universal coefficient theorem, and the left-

hand map uses the standard presentation for H∗(BU (n);Z), along with our grading convention |ci | −2i.

Now we proceed to the proof of the main theorem.

Proof of Theorem 2.4.1. By definition,

H∗(

(Holst (BU (n)))−TBU (n))

inv limmH∗((

i∗mHol(BU (n)))−TGr(n ,m)

)

and by Lemma 2.3.1,

inv limmH∗(


inv limm (H−∗(Gr(n ,m); H∗(Rat(BU (n))))

But by Lemma 2.4.2 and Lemma 2.4.3, the canonical morphism

H∗(Rat(BU (n)))[[c1 , . . . , cn]] −→ inv limm (H−∗(Gr(n ,m); H∗(Rat(BU (n))))

is an isomorphism of graded rings, where the grading on the right-hand side is derived from the Cohen-Jones-

Yan spectral sequence, and the grading on the left-hand side is given by setting |ci | −2i, and then using the

usual grading for a tensor product of graded rings. This proves the theorem.

Chapter 3

Cohomology, tableaux, and homology

operations

In this chapter we give more information about the pro-spectrum Holst (BU (n))−TBU (n) , beyond the integral

homology which was completely described in Chapter 2. In Section 3.1 we describe the ind-system obtained

from applying the integral cohomology functor H∗(−;Z) to the pro-spectrum Holst (BU (n))−TBU (n) . It

is important to keep in mind here that there is no simple duality relationship between inv lim H∗(X•) and

dir lim H∗(X•) if X• is a pro-spectrum.

3.1 Cohomology of holomorphic curves in BU (n)

3.1.1 Young tableaux

For what follows, we will use the description of the cohomology of complex Grassmannians in terms of

Young tableaux, so we will quickly review this theory.

Let r ≤ nm and let

λ (λ1 , . . . , λn )

33

CHAPTER 3. COHOMOLOGY, TABLEAUX, AND HOMOLOGY OPERATIONS 34

to be an integer partition of r, i.e.n∑

i1

λi r

which satisfies

m ≥ λ1 ≥ λ2 ≥ . . . ≥ λn ≥ 0.

Fix a complete flag in Cn+m , i.e. a sequence of linear subspaces

0 V0 ( V1 ( . . . ( Vn+m Cn+m ,

and for a given partition λ as above, define the Schubert cell σλ to be

σλ

W ∈ Gr(n ,m) |W ∩ Vm+i−λi i ,W ∩ Vm+i−λi−1 i − 1

⊂ Gr(n ,m)

Then σλ is a 2r-dimensional cell in Gr(n ,m) and we have [σλ] ∈ H2r (Gr(n ,m);Z). In fact, the set of

[σλ] for λ satisfying the above constraints form a Z-basis for H2r (Gr(n ,m);Z). We will denote elements

of the dual basis for H2r (Gr(n ,m);Z) by [σλ]∗.

Proposition 3.1.1. The map induced by inclusion

i∗ : H∗(Gr(n ,m)) −→ H∗(Gr(n ,m + 1)),

on the level of Young tableaux, is simply the identity.

Proof. If

0 V0 ( V1 ( . . . ( Vn+m+1 Cn+m+1 ,

is a complete flag in Cn+m+1, then by truncating Vn+m+1 we obtain a complete flag in Cn+m . Then if W ⊆

Cn+m is a subspace, and i : Cn+m → Cn+m+1 is the inclusion, then

dim W ∩ Vk dim i(W ) ∩ Vk

for all k.


We will need the following three facts.

Proposition 3.1.2. ([19], 3.5.5) The kth Chern class in homology ck ∈ H2k (Gr(n ,m)) is identical to [σλ]

where λ is a row of k boxes

· · ·︸︷︷︸

k

Proof.

Proposition 3.1.3. Let [σλ] ∈ H2r (Gr(n ,m)). Then the Poincaré dual of [σλ] ∈ H2(nm−r) (Gr(n ,m)) is

[σλ′]∗, where the Young tableau λ′ is defined to be the complement of λ in an m × n array.

Example. The complement of λ (4, 2, 1, 0) ⊂ 4 × 4 is λ′ (4, 3, 2, 0).

Lemma 3.1.4. (Pieri’s formula, [19] 3.2.8) If λ is a partition with |λ | r, and 1 ≤ k ≤ n, then

[σλ]∗ ∪ ck

∑

nu

σν ∈ H2(r+k) (Gr(n ,m))

where the sum is taken over partitions ν which are obtained from λ by adding k boxes, no two in the same

column.

3.1.2 Maps in the Cohomology Ind-System

The following could be regarded as a complement to Theorem 2.3.3, as mentioned in the chapter introduction.

Proposition 3.1.5. There is a commutative diagram

H∗(Gr(n ,m)−TGr(n ,m)) //

∪τm

H∗(Gr(n ,m + 1)−TGr(n ,m+1)

∪τm−1

H∗+2mn (Gr(n ,m)) // H∗+2(m+1)n (Gr(n ,m + 1))

Proof. By Lemma 1.1.9, it suffices to identify (i!)∗ with ∪cn . We will do this using the language of Young

tableaux, and it suffices to prove that the maps agree on basis elements [σλ]∗ for λ ⊆ m × n a partition.

The following diagram commutes.


H∗(Gr(n ,m))(i!)∗

//

P.D.

H∗(Gr(n ,m + 1))

P.D.

H2mn−∗(Gr(n ,m))i∗

// H2(m+1)n−∗(Gr(n ,m + 1)))

Therefore, (i!)∗ is defined on basis elements by

[σλ]∗ 7→ [σλ′′]∗

[σλ]∗ by taking the complement tableaux λ′, including it in the larger array (m + 1) × n, and then taking the

complement again, λ′′. By an easy combinatorial argument, λ′′ is obtained from λ by adding a single box in

every column.

On the other hand, by Lemma 3.1.4, Pieri’s formula,

[σλ]∗ ∪ cn

∑

ν

[σν]∗

where ν is obtained from λ by adding n boxes; at most one in each column. But, because λ lives in m × n,

there is only one element in this sum, which is [σλ′′]∗.

Therefore, (i!)∗ agrees with ∪cn on basis elements [σλ]∗, and we may conclude that the two maps are

identical.

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